DOCSLIB.ORG

Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Geometry of the projectivization of ideals and applications to problems of birationality Remi Bignalet-Cazalet To cite this version: Remi Bignalet-Cazalet. Geometry of the projectivization of ideals and applications to problems of birationality. Algebraic Geometry [math.AG]. Université Bourgogne Franche-Comté, 2018. English. NNT : 2018UBFCK038. tel-02004276 HAL Id: tel-02004276 https://tel.archives-ouvertes.fr/tel-02004276 Submitted on 1 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` de doctorat de l'Universite´ de Bourgogne Franche-Comte´ pr´epar´ee`al'Institut de Math´ematiquesde Bourgogne Ecole´ doctorale Carnot Pasteur (ED553) Doctorat de Math´ematiques par R´emiBignalet-Cazalet Geom´ etrie´ de la projectivisation des ideaux´ et applications aux problemes` de birationalite´ Directeurs de th`ese: Adrien Dubouloz et Daniele Faenzi Th`esesoutenue le 24 octobre 2018 `aDijon Composition du jury Laurent Buse´ INRIA - Sophia Antipolis Examinateur Julie Deserti´ IMJ-PRG - Universit´eParis Diderot Examinatrice Adrien Dubouloz IMB - Universit´ede Bourgogne Directeur de th`ese Daniele Faenzi IMB - Universit´ede Bourgogne Directeur de th`ese Laurent Manivel IMT - Universit´ePaul Sabatier Rapporteur Lucy Moser-Jauslin IMB - Universit´ede Bourgogne Pr´esidente du jury Fransesco Russo DMI - Universit`adegli Studi di Catania Rapporteur Ronan Terpereau IMB - Universit´ede Bourgogne Examinateur Institut de Math´ematiquesde ´ Bourgogne Ecole Doctorale Carnot-Pasteur UMR 5584 CNRS Universit´ede Bourgogne Universit´ede Bourgogne Franche-Comt´e Franche-Comt´e UFR Sciences et Techniques 9 Avenue Alain Savary 9 Avenue Alain Savary BP 47870 21078 Dijon Cedex BP 47870 21078 Dijon Cedex 1 Al andar se hace camino y al volver la vista atr´as se ve la senda que nunca se ha de volver a pisar. Antonio Machado, Caminante no hay camino 2 3 En marchant se fait le chemin et c'est en se retournant que l'on peut contempler le sentier que l'on n'aura jamais plus l'occasion d'emprunter. Traduction personnelle 4 5 Remerciements Au cours de ma th`ese,j'ai pu b´en´eficierdu concours, du soutien ou de l'aide de personnes que je tiens `aremercier. J'esp`ereque ces quelques lignes traduiront ma profonde gratitude envers eux. Mes premiers remerciements vont `ames deux directeurs de th`ese, Adrien Dubouloz et Daniele Faenzi. Que de patience, que d'indulgence de leur part durant ces trois ann´ees! Je tiens `ales remercier particuli`erement pour leur disponibilit´e et leur exigence lors de toutes les phases r´edactionnellesde mon doctorat. J'ai pu comprendre, gr^ace`aeux, les passages oblig´esd'´elaboration, de maturation et de diffusion d'une id´eemath´ematique.J'ai beaucoup appr´eci´eaussi les discussions passionn´eesd'Adrien sur le fonctionnement du monde math´ematiquequ'il m'a aid´e `amieux envisager et la clairvoyance et les fulgurances de Daniele qui ont ´et´edes sources constantes d'inspiration. Un grand merci aussi `a • Laurent Manivel et Francesco Russo pour avoir rapport´ecette th`ese.Leurs remarques ont permis de grandement am´eliorerle manuscrit initial et m'ont fait prendre conscience de questions que je n'avais pas encore clairement identifi´ees,notamment dans les chapitres 4 et 6, • Julie D´esertiqui m'a aiguill´esur cette th`eseapr`esavoir encadr´emon m´emoire de master. Les th´ematiquesque j'ai abord´eesdurant ces trois ann´eesont ´et´e tr`esinfluenc´eespar ce travail initial et je la remercie d'autant plus d'avoir accept´ed'^etredans mon jury de th`ese. • Laurent Bus´e,Lucy Moser-Jauslin et Ronan Terpereau pour avoir accept´e de faire partie de mon jury de th`eseet particuli`erement Ronan dont j'ai ´enorm´ement pu appr´ecierle dynamisme et l'engagement dans les activit´es math´ematiqueset extra-math´ematiquesdijonnaises. De mani`ereg´en´erale,merci `ala dynamique ´equipe de G´eom´etriealg´ebrique de Dijon, dont Fr´ed´ericD´egliseet Jan Nagel, pour m'avoir fait d´ecouvriret associ´e `atant d'activit´esmath´ematiques,`aJos´e-Luispour sa bienveillance, `aEmmanuel qui m'a mis le pied `al'´etrierde l'enseignement `al'universit´eainsi qu’`aPeggy et Fran¸coispour leur aide pr´ecieusedans certaines de mes d´emarches administratives. Je tiens `aremercier Charlie dont la fiabilit´er´esiste `atoute ´epreuve et ceux qui m'ont rendu la vie de doctorant `aDijon plus agr´eable,les membres de l'IMB et toute son ´equipe administrative, les doctorants ou ex-doctorants de l'IMB, en particulier mes deux anciens co-brasseurs Antoine et J´er´emy avec qui j'ai pass´eune premi`ere ann´eede th`eseparticuli`erement productive... Je tiens `aciter Matilde, Vladimiro, Andr`es,Anne, Mathias, Fabio dont j'ai ´enorm´ement appr´eci´ela compagnie entre autre dans des conf´erencesou ´ecolesde maths. Un merci plus sp´ecial `aMatilde pour m'avoir accueilli dans cette belle ville de San Marin durant quelques jours si particuliers d'ao^ut.Plus g´en´eralement, je remercie tous mes amis, palois, palois d´esormaisparisiens et autres provinciaux parisiens, dijonnais ou anciens dijonnais ainsi que toute ma famille qui ont ´et´ed'un soutien sans faille tout au long de mon doctorat. Dijon, le 10 octobre 2018. 6 R´esum´egrand public L'explosion r´ecente des capacit´esde calculs num´eriquesqui s'est produite au cours des trente derni`eres ann´eesa renvers´eles perspectives de recherches et d´eveloppements. La simulation num´eriquepermet d´esormaisune estima- tion plus fiable et beaucoup plus rapide de r´esultatsautrefois inatteignables. Dans cette th`ese,nous appliquons ce principe tr`esg´en´eral`aun domaine des math´ematiques,la g´eom´etriealg´ebrique,qui concerne l'´etudedes ´equations polynomiales. Cela nous permet de pr´evoir l'existence de configurations g´eom´etriquesinattendues apportant ainsi un point de vue original sur des su- jets classiques de g´eom´etrie.En plus de ces simulations, nous nous attachons `ad´emontrer math´ematiquement ces ph´enom`enesannonc´esnum´eriquement. Nos r´esultatsconcernent aussi le d´eveloppement de m´ethodes num´eriques visant `aam´eliorerdavantage les capacit´eset la rapidit´edes simulations dans ce domaine. Large audience abstract The recent growth in capacity of numerical calculus over the last thirty years has brought a major change in perspective regarding matters of re- search and development. Numerical simulation allows a more reliable and faster estimation of results unattainable beforehand. In this thesis, we apply this very general principle to a domain of mathematics, algebraic geometry which concerns the study of polynomial equations. This allows us to predict unexpected geometrical configurations bringing an original point of view to classical geometrical subjects. In addition of those simulations, we focus on proving mathematically those predicted phenomenons. Our results also con- cern the development of numerical methods aiming to further improve the capacity and the rapidity of simulations in this area. 7 R´esum´e Dans cette th`ese,nous interpr´etonsg´eom´etriquement la torsion de l'alg`ebre sym´etriqued'un faisceau d'id´eaux IZ d'un sch´ema Z d´efinipar n+1 ´equations dans une vari´et´en-dimensionnelle. Ceci revient `a´etudierla g´eom´etriede la projectivisation de IZ . Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l'espace projectif de dimension 3 au sujet duquel nous construisons des transformations bi- rationnelles explicites qui ont le m^emedegr´ealg´ebrique que leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous g´en´eralisons une caract´erisationdes courbes libres en ´etendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersur- faces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degr´e5 en caract´eristique3. La derni`ere application concerne le calcul de l'inverse d'une transformation birationnelle. Mots cl´es: G´eom´etriealg´ebrique,Alg`ebrecommutative, Th´eoriedes singularit´es, Transformations birationelles, Hypersurfaces homalo¨ıdes,courbes libres et presque libres, alg`ebrede Rees et alg`ebresymm´etrique,Syzygies, R´esolutions Title of the thesis : Geometry of the projectivization of ideals and applications to problems of birationality Abstract In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf IZ of a scheme Z defined by n + 1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of IZ . The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map. Keywords: Algebraic Geometry, Commutative algebra, Singularity theory, Bira- tional maps, Homaloidal hypersurfaces, free and nearly free curves, Symmetric and Rees algebra, Syzygies, Resolutions 8 Contents Introduction 11 Contents of the manuscript .......................... 17 1 Degrees of rational maps 23 1.1 Proj of a sheaf .............................
Recommended publications
  • ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme

    ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme

    ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ).
  • SHEAVES of MODULES 01AC Contents 1. Introduction 1 2

    SHEAVES of MODULES 01AC Contents 1. Introduction 1 2

    SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter.
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space

    4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space

    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
  • Essential Concepts of Projective Geomtry

    Essential Concepts of Projective Geomtry

    Essential Concepts of Projective Geomtry Course Notes, MA 561 Purdue University August, 1973 Corrected and supplemented, August, 1978 Reprinted and revised, 2007 Department of Mathematics University of California, Riverside 2007 Table of Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i Prerequisites: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :iv Suggestions for using these notes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :v I. Synthetic and analytic geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :1 1. Axioms for Euclidean geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Cartesian coordinate interpretations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 3 3. Lines and planes in R and R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 II. Affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Synthetic affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2. Affine subspaces of vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3. Affine bases: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :19 4. Properties of coordinate
  • Math.AG] 28 Feb 2007 Osnua Leri Variety Algebraic Nonsingular a the Se[A) Let [Ma])

    Math.AG] 28 Feb 2007 Osnua Leri Variety Algebraic Nonsingular a the Se[A) Let [Ma])

    SOME BASIC RESULTS ON ACTIONS OF NON-AFFINE ALGEBRAIC GROUPS MICHEL BRION Abstract. We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. 0. Introduction Algebraic group actions have been extensively studied under the as- sumption that the acting group is affine or, equivalently, linear; see [KSS, MFK, PV]. In contrast, little seems to be known about actions of non-affine algebraic groups. In this paper, we show that these actions may be reduced to actions of affine subgroup schemes, in the setting of normal varieties. Our starting point is the following theorem of Nishi and Matsumura (see [Ma]). Let G be a connected algebraic group of automorphisms of a nonsingular algebraic variety X and denote by αX : X −→ A(X) the Albanese morphism, that is, the universal morphism to an abelian variety (see [Se2]). Then G acts on A(X) by translations, compatibly with its action on X, and the kernel of the induced homomorphism G → A(X) is affine. When applied to the action of G on itself via left multiplication, this shows that the Albanese morphism αG : G −→ A(G) is a surjective group homomorphism having an affine kernel. Since arXiv:math/0702518v2 [math.AG] 28 Feb 2007 this kernel is easily seen to be smooth and connected, this gives back Chevalley’s structure theorem: any connected algebraic group G is an extension of an abelian variety A(G) by a connected affine algebraic group Gaff (see [Co]) for a modern proof). The Nishi–Matsumura theorem may be reformulated as follows: for any faithful action of G on a nonsingular variety X, the induced homo- morphism G → A(X) factors through a homomorphism A(G) → A(X) having a finite kernel (see [Ma] again).
  • Generalized Divisors and Reflexive Sheaves

    Generalized Divisors and Reflexive Sheaves

    GENERALIZED DIVISORS AND REFLEXIVE SHEAVES KARL SCHWEDE 1. Preliminary Commutative Algebra We introduce the notions of depth and of when a local ring (or module over a local ring) is \S2". These notions are found in most books on commutative algebra, see for example [Mat89, Section 16] or [Eis95, Section 18]. Another excellent book that is focussed on these ideas is [BH93]. We won't be focussing on the commutative algebra, but one should be aware, at the very least, that this background exists. In particular, we won't really use any of the theory on Cohen-Macaulay rings. However, since one ought to build up the same machinery in order to define S2, we include these definitions as well. All rings will be assumed to be noetherian. Definition 1.1. Let (R; m) be a local ring and let M be a finite R-module (which means finitely generated). An element r 2 R is said to be M-regular if rx 6= 0 for all x 2 M, x 6= 0 (in other words, r is not a zero divisor on M). A sequence of elements r1; : : : ; rn 2 R is said to be M-regular if (i) ri is a regular M=(r1; : : : ; ri−1) element for all i ≥ 1 (in particular r1 is M-regular) (ii) (r1; : : : ; rn)M ( M (that is, the containment is proper). Remark 1.2. If condition (i) is satisfied, but condition (ii) is not necessarily satisfied then such a sequence is called weakly M-regular. Remark 1.3. Notice that condition (ii) implies in a regular sequence, all ri 2 m 2 R.
  • 4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves

    4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves

    4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module.
  • AG for NT Week 6

    AG for NT Week 6

    AG for NT Week 6 Recap 1. A ring, X = Spec A, M an A-module which we associated a sheaf Mf of OX -modules. Have shown: Any quasi-coherent sheafs on Spec A has this form. 2. S graded ring, X = PrjS, M graded S-module, associated to it Mf a quasi-coherent sheaf. Mf(D+(f)) = fhomogeneous elements of degree 0 in Mf g. Want to show: Given any quasi-coherent sheaf on X = ProjS, we have f =∼ Mf for some graded S-module M. We won't (can't?) say this in general, but Proposition 0.1. S graded ring, nitely generated by S1 as an S0-algebra (S = ⊕a≥0Sa). Let X = ProjS and f a quasi coherent sheaf on X. Then there is a graded S-module M such that f =∼ Mf. Remark. 1. Applies to for some ring , i.e., to n S = A[x0; : : : ; xn] A PA 2. Don't have an equivalence of categories between graded S-modules and quasi-coherent sheafs on ProjS. Indeed, 2 dierent graded modules can give the same sheaf. Example: and for some ⊕n≥0Mn ⊕n≥n0 Mn n0 ≥ 0 3. How to dene M given f? Cannot just set M = Γ(X; f) as in the ane case. Solution is to use twists. Example. , Proj r . S = A[x0; : : : ; xr] X = S = PA Claim. Γ(X; OX (n)) = Sn = fhomogenous polynomials of degree n}. In ∼ particular, Γ(X; OX ) = A. Proof. The sets D+(xi) cover X. A section t 2 Γ(X; OX (n)) is the same as an (r + 1)-tuple of sections (t0; : : : ; tr) with ti 2 OX (n)(D+(xi)) com- patible on D+(xi) \ D+(xj) = D+(xixj) Now, homogeneous elements in and OX (n)(D+(xi)) = fdeg n Sxi g OX (n)(D+(xixj)) = homogeneous elements in .
  • Notes on Automorphism Groups of Projective Varieties

    Notes on Automorphism Groups of Projective Varieties

    NOTES ON AUTOMORPHISM GROUPS OF PROJECTIVE VARIETIES MICHEL BRION Abstract. These are extended and slightly updated notes for my lectures at the School and Workshop on Varieties and Group Actions (Warsaw, September 23{29, 2018). They present old and new results on automorphism groups of normal projective varieties over an algebraically closed field. Contents 1. Introduction 1 2. Some basic constructions and results 4 2.1. The automorphism group 4 2.2. The Picard variety 7 2.3. The lifting group 10 2.4. Automorphisms of fibrations 14 2.5. Big line bundles 16 3. Proof of Theorem 1 18 4. Proof of Theorem 2 20 5. Proof of Theorem 3 23 References 28 1. Introduction Let X be a projective variety over an algebraically closed field k. It is known that the automorphism group, Aut(X), has a natural structure of smooth k-group scheme, locally of finite type (see [Gro61, Ram64, MO67]). This yields an exact sequence 0 (1.0.1) 1 −! Aut (X) −! Aut(X) −! π0 Aut(X) −! 1; where Aut0(X) is (the group of k-rational points of) a smooth connected algebraic group, and π0 Aut(X) is a discrete group. To analyze the structure of Aut(X), one may start by considering the connected automorphism 0 group Aut (X) and the group of components π0 Aut(X) separately. It turns out that there is no restriction on the former: every smooth connected algebraic group is the connected automorphism group of some normal projective variety X (see [Bri14, Thm. 1]). In characteristic 0, we may further take X to be smooth by using equivariant resolution of singularities (see e.g.
  • HOMOLOGICAL PROJECTIVE DUALITY � by ALEXANDER KUZNETSOV

    HOMOLOGICAL PROJECTIVE DUALITY by ALEXANDER KUZNETSOV

    HOMOLOGICAL PROJECTIVE DUALITY by ALEXANDER KUZNETSOV ABSTRACT We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singu- larities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles. 1. Introduction Investigation of derived categories of coherent sheaves on algebraic varieties has become one of the most important topics in the modern algebraic geometry. Among other reasons, this is because of the Homological Mirror Symmetry con- jecture of Maxim Kontsevich [Ko] predicting that there is an equivalence of cat- egories between the derived category of coherent sheaves on a Calabi–Yau variety and the derived Fukaya category of its mirror. There is an extension of Mirror Symmetry to the non Calabi–Yau case [HV]. According to this, the mirror of a manifold with non-negative first Chern class is a so-called Landau–Ginzburg model, that is an algebraic variety with a 2-form and a holomorphic function (superpotential) such that the restriction of the 2-form to smooth fibers of the superpotential is symplectic. It is expected that singular fibers of the superpoten- tial of the mirror Landau–Ginzburg model give a decomposition of the derived category of coherent sheaves on the initial algebraic variety into semiorthogonal pieces, a semiorthogonal decomposition.
  • K-Stability of Relative Flag Varieties

    K-Stability of Relative Flag Varieties

    K-stability of relative flag varieties Anton Isopoussu University of Cambridge Department of Pure Mathematics and Mathematical Statistics Churchill College arXiv:1307.7638v2 [math.AG] 10 Nov 2015 April 2015 This dissertation is submitted for the degree of Doctor of Philosophy Abstract We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The strongest results are obtained when working over a Riemann surface. Weaker partial results are obtained for adiabatic polarisations in the general case. We develop the notion of relative K-stability which embeds the idea of working over a base variety into the theory of K-stability. We equip the set of equivalence classes of test configuration with the structure of a convex space fibred over the cone of rational polarisations. From this, we deduce the open- ness of the K-unstable locus. We illustrate our new algebraic constructions with several examples. Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University of similar institution except as specified in the text Anton Isopoussu November 11, 2015 Acknowledgements I gratefully acknowledge the patient guidance of my supervisor Dr.
  • GEOMETRIC METHODS in COMMUTATIVE ALGEBRA 1. Local

    GEOMETRIC METHODS in COMMUTATIVE ALGEBRA 1. Local

    GEOMETRIC METHODS IN COMMUTATIVE ALGEBRA STEVEN DALE CUTKOSKY 1. Local Cohomology Suppose that R is a Noetherian ring, and M is an R-module. Suppose that I is an ideal in R, with generators I = (f1; : : : ; fn). Consider the modified Cech complex C∗ : 0 ! C0 ! C1 !···! Cd ! 0 Where C0 = R and M Ct = R : fi1 fi2 ···fit 1≤i1<i2<···<it≤n The Local Cohomology of M is i i ∗ HI (M) = H (M ⊗R C ): 0 k HI (M) = ff 2 M j I f = 0 for some k ≥ 0g = ΓI (M); the set of elements of M which have support in I. i HI (M) does not depend on the choice of generators of I. Further, Hi(M) = Hpi (M): I I ΓI (∗) is a left exact functor, and the cohomology modules are its right derived functors. If 0 ! A ! B ! C ! 0 is a short exact sequence of R-modules, then there is a long exact sequence 0 0 0 1 0 ! HI (A) ! HI (B) ! HI (C) ! HI (A) ! ::: 1.1. Sheaf Cohomology. Let X = spec(R). Let M = M~ be the quasi coherent sheaf on X associated to M. M~ is determined by the \stalks" (M~ )p = Mp for all p 2 X. We have that for f 2 R, the \sections" over the open set D(f) = X n V (f) are Γ(D(f); M) = Mf : n From the sheaf axioms, and since [i=1D(fi) = U, we have that Γ(U; M), where U = X n V (I), is computed as the kernel of d0: n M d0 M Γ(D(fi); M) ! Γ(D(fj) \ D(fk); M): i=1 j<k This map is just n M d0 M Mfi ! Mfj fk : i=1 j<k Consider the Cech complex F ∗ : F 0 ! F 1 !···! F n−1; 1 where M F t = R : fi1 fi2 ···fit+1 1≤i1<i2<···<it+1≤n The sheaf cohomology of M~ on U = X n V (I) is i i ∗ H (U; M~ ) = H (M ⊗R F ): We have that H0(U; M~ ) = Γ(U; M~ ); the sections of M~ over U = X n V (I).